pacman::p_load(sf, raster, spatstat, tmap, tidyverse, rvest, geojsonsf)Spatial Point Patterns Analysis
1.0 Context:
Spatial Point Pattern Analysis involves evaluating the pattern or distribution of a set of points on a surface. These points can represent the locations of:
Events, such as crimes, traffic accidents, or disease outbreaks, or
Business services, like coffee shops and fast food outlets, or facilities such as childcare and eldercare centers.
The specific questions we would like to answer are as follows:
are the childcare centres in Singapore randomly distributed throughout the country?
if the answer is not, then the next logical question is where are the locations with higher concentration of childcare centres?
2.0 Downloading the Data setsss
To provide answers to the questions above, three data sets will be used. They are:
CHILDCARE, a point feature data providing both location and attribute information of childcare centres. It was downloaded from Data.gov.sg and is in geojson format. Link hereMP14_SUBZONE_WEB_PL, a polygon feature data providing information of URA 2014 Master Plan Planning Subzone boundary data. It is in ESRI shapefile format. This data set was also downloaded from Data.gov.sg. Link hereCostalOutline, a polygon feature data showing the national boundary of Singapore. It is provided by SLA and is in ESRI shapefile format. Link here
3.0 Installing and loading R packages
In this hands-on exercise, five R packages will be used, they are:
sf, a relatively new R package specially designed to import, manage and process vector-based geospatial data in R.
spatstat, which has a wide range of useful functions for point pattern analysis. In this hands-on exercise, it will be used to perform 1st- and 2nd-order spatial point patterns analysis and derive kernel density estimation (KDE) layer.
raster which reads, writes, manipulates, analyses and model of gridded spatial data (i.e. raster). In this hands-on exercise, it will be used to convert image output generate by spatstat into raster format.
maptools which provides a set of tools for manipulating geographic data. In this hands-on exercise, we mainly use it to convert Spatial objects into ppp format of spatstat.
tmap which provides functions for plotting cartographic quality static point patterns maps or interactive maps by using leaflet API.
Use the code chunk below to install and launch the five R packages.
4.0 Spatial Data Wrangling
4.1 Importing Spatial Data
childcare_sf <- st_read("data/child-care-services-geojson.geojson") %>%
st_transform(crs = 3414)Reading layer `child-care-services-geojson' from data source
`C:\Users\jiale\Desktop\IS415\IS415-GAA\Hands_On_Exercises\Hands_On_Exercise_3\data\child-care-services-geojson.geojson'
using driver `GeoJSON'
Simple feature collection with 1545 features and 2 fields
Geometry type: POINT
Dimension: XYZ
Bounding box: xmin: 103.6824 ymin: 1.248403 xmax: 103.9897 ymax: 1.462134
z_range: zmin: 0 zmax: 0
Geodetic CRS: WGS 84sg_sf <- st_read(dsn = "data", layer="CostalOutline")Reading layer `CostalOutline' from data source
`C:\Users\jiale\Desktop\IS415\IS415-GAA\Hands_On_Exercises\Hands_On_Exercise_3\data'
using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension: XY
Bounding box: xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21mpsz_sf <- st_read(dsn = "data", layer = "MP14_SUBZONE_WEB_PL")Reading layer `MP14_SUBZONE_WEB_PL' from data source
`C:\Users\jiale\Desktop\IS415\IS415-GAA\Hands_On_Exercises\Hands_On_Exercise_3\data'
using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY214.1.1 DIY:Use the appropriate SF function to retrieve the referencing system information of these geospatial data.
Simple, use the st_crs function from SF to check and print the crs information
# Retrieve CRS information
childcare_crs <- st_crs(childcare_sf)
sg_crs <- st_crs(sg_sf)
mpsz_crs <- st_crs(mpsz_sf)
# Print CRS information
print(childcare_crs)Coordinate Reference System:
User input: EPSG:3414
wkt:
PROJCRS["SVY21 / Singapore TM",
BASEGEOGCRS["SVY21",
DATUM["SVY21",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4757]],
CONVERSION["Singapore Transverse Mercator",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",1.36666666666667,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",103.833333333333,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",1,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",28001.642,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",38744.572,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["northing (N)",north,
ORDER[1],
LENGTHUNIT["metre",1]],
AXIS["easting (E)",east,
ORDER[2],
LENGTHUNIT["metre",1]],
USAGE[
SCOPE["Cadastre, engineering survey, topographic mapping."],
AREA["Singapore - onshore and offshore."],
BBOX[1.13,103.59,1.47,104.07]],
ID["EPSG",3414]]print(sg_crs)Coordinate Reference System:
User input: SVY21
wkt:
PROJCRS["SVY21",
BASEGEOGCRS["SVY21[WGS84]",
DATUM["World Geodetic System 1984",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]],
ID["EPSG",6326]],
PRIMEM["Greenwich",0,
ANGLEUNIT["Degree",0.0174532925199433]]],
CONVERSION["unnamed",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",1.36666666666667,
ANGLEUNIT["Degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",103.833333333333,
ANGLEUNIT["Degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",1,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",28001.642,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",38744.572,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["(E)",east,
ORDER[1],
LENGTHUNIT["metre",1,
ID["EPSG",9001]]],
AXIS["(N)",north,
ORDER[2],
LENGTHUNIT["metre",1,
ID["EPSG",9001]]]]print(mpsz_crs)Coordinate Reference System:
User input: SVY21
wkt:
PROJCRS["SVY21",
BASEGEOGCRS["SVY21[WGS84]",
DATUM["World Geodetic System 1984",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]],
ID["EPSG",6326]],
PRIMEM["Greenwich",0,
ANGLEUNIT["Degree",0.0174532925199433]]],
CONVERSION["unnamed",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",1.36666666666667,
ANGLEUNIT["Degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",103.833333333333,
ANGLEUNIT["Degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",1,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",28001.642,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",38744.572,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["(E)",east,
ORDER[1],
LENGTHUNIT["metre",1,
ID["EPSG",9001]]],
AXIS["(N)",north,
ORDER[2],
LENGTHUNIT["metre",1,
ID["EPSG",9001]]]]4.1.2 DIY: Assign the correct CRS to MPSZ_SF and SG_SF Simple Feature Data frames.
notice that the MPSZ_SF and SG_SF is in World Geodetic System 1984 format, we need set the correct crs to these data and we can do so using the st transform. We can do so using the transform method
mpsz_sf <- st_read(dsn = "data", layer = "MP14_SUBZONE_WEB_PL") %>%
st_transform(crs = 3414)Reading layer `MP14_SUBZONE_WEB_PL' from data source
`C:\Users\jiale\Desktop\IS415\IS415-GAA\Hands_On_Exercises\Hands_On_Exercise_3\data'
using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21sg_sf <- st_read(dsn = "data", layer = "CostalOutline") %>%
st_transform(crs = 3414)Reading layer `CostalOutline' from data source
`C:\Users\jiale\Desktop\IS415\IS415-GAA\Hands_On_Exercises\Hands_On_Exercise_3\data'
using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension: XY
Bounding box: xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21print(st_crs(mpsz_sf))Coordinate Reference System:
User input: EPSG:3414
wkt:
PROJCRS["SVY21 / Singapore TM",
BASEGEOGCRS["SVY21",
DATUM["SVY21",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4757]],
CONVERSION["Singapore Transverse Mercator",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",1.36666666666667,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",103.833333333333,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",1,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",28001.642,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",38744.572,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["northing (N)",north,
ORDER[1],
LENGTHUNIT["metre",1]],
AXIS["easting (E)",east,
ORDER[2],
LENGTHUNIT["metre",1]],
USAGE[
SCOPE["Cadastre, engineering survey, topographic mapping."],
AREA["Singapore - onshore and offshore."],
BBOX[1.13,103.59,1.47,104.07]],
ID["EPSG",3414]]print(st_crs(sg_sf))Coordinate Reference System:
User input: EPSG:3414
wkt:
PROJCRS["SVY21 / Singapore TM",
BASEGEOGCRS["SVY21",
DATUM["SVY21",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4757]],
CONVERSION["Singapore Transverse Mercator",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",1.36666666666667,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",103.833333333333,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",1,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",28001.642,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",38744.572,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["northing (N)",north,
ORDER[1],
LENGTHUNIT["metre",1]],
AXIS["easting (E)",east,
ORDER[2],
LENGTHUNIT["metre",1]],
USAGE[
SCOPE["Cadastre, engineering survey, topographic mapping."],
AREA["Singapore - onshore and offshore."],
BBOX[1.13,103.59,1.47,104.07]],
ID["EPSG",3414]]4.1.3 Change the referencing System to Singapore National Projected Coordinate System
Note
Understanding the CRS in Our Data:
MPZ and Coastal Data:
CRS:
SVY21, which is the Singapore National Projected Coordinate System based on WGS84.Description: This is a common projected coordinate system used in Singapore for accurate mapping.
Childcare Data:
CRS:
SVY21 / Singapore TM(Transverse Mercator projection).Description: This is also a projection based on SVY21, specifically using the Transverse Mercator projection. It is very close to the SVY21 system, with minor differences in how the projection is handled.
Given that the map file serves as the base, we want all spatial data to overlay correctly, we should:
Transform the GeoJSON Data to Match the Map File’s CRS:
- Since our MPZ and Coastal data are already in
SVY21(EPSG:3414), transform the GeoJSON data toEPSG:3414as well.
- Since our MPZ and Coastal data are already in
Rationale:
This approach ensures that the childcare locations from the GeoJSON data will be accurately plotted within the boundaries and context provided by the map file (MPZ and Coastal data).
It avoids potential issues with misalignment, especially since oour base map data is already set up in a local projection suitable for Singapore.
# Transform Childcare data to match the base map's CRS (EPSG:3414)
childcare_sf <- st_read("data/child-care-services-geojson.geojson") %>%
st_transform(crs = 3414)Reading layer `child-care-services-geojson' from data source
`C:\Users\jiale\Desktop\IS415\IS415-GAA\Hands_On_Exercises\Hands_On_Exercise_3\data\child-care-services-geojson.geojson'
using driver `GeoJSON'
Simple feature collection with 1545 features and 2 fields
Geometry type: POINT
Dimension: XYZ
Bounding box: xmin: 103.6824 ymin: 1.248403 xmax: 103.9897 ymax: 1.462134
z_range: zmin: 0 zmax: 0
Geodetic CRS: WGS 84# Now, all datasets should be aligned in the same CRS4.1.4 Checking for validity of maps
When working with spatial data, it’s crucial to ensure that all geometries are valid. Invalid geometries can cause errors in analysis and visualization.
- Checking Validity with
st_is_valid(): - Identifying Invalid Geometries:
- Fixing Invalid Geometries with
st_make_valid()
mpsz_validity <- st_is_valid(mpsz_sf)
mpsz_invalid <- which(!mpsz_validity)
if (length(mpsz_invalid) > 0) {
print("MPZ Invalid!")
print(mpsz_sf[mpsz_invalid, ])
} else {
print("it's valid!")
}[1] "MPZ Invalid!"
Simple feature collection with 9 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: 12535.88 ymin: 21678.35 xmax: 56396.44 ymax: 49291.03
Projected CRS: SVY21 / Singapore TM
OBJECTID SUBZONE_NO SUBZONE_N SUBZONE_C CA_IND
19 19 2 SOUTHERN GROUP SISZ02 N
20 20 1 SENTOSA SISZ01 N
24 24 1 MARITIME SQUARE BMSZ01 N
122 122 9 JURONG PORT JESZ09 N
123 123 3 SAMULUN BLSZ03 N
128 128 9 PANDAN CLSZ09 N
258 258 4 PASIR RIS PARK PRSZ04 N
302 302 1 NORTH-EASTERN ISLANDS NESZ01 N
320 320 9 NORTH COAST WDSZ09 N
PLN_AREA_N PLN_AREA_C REGION_N REGION_C
19 SOUTHERN ISLANDS SI CENTRAL REGION CR
20 SOUTHERN ISLANDS SI CENTRAL REGION CR
24 BUKIT MERAH BM CENTRAL REGION CR
122 JURONG EAST JE WEST REGION WR
123 BOON LAY BL WEST REGION WR
128 CLEMENTI CL WEST REGION WR
258 PASIR RIS PR EAST REGION ER
302 NORTH-EASTERN ISLANDS NE NORTH-EAST REGION NER
320 WOODLANDS WD NORTH REGION NR
INC_CRC FMEL_UPD_D X_ADDR Y_ADDR SHAPE_Leng SHAPE_Area
19 5809FC547293EA2D 2014-12-05 29815.09 23412.59 25626.977 2206319
20 A6FCDC9C447952CB 2014-12-05 27593.94 25813.35 17496.194 4919132
24 C1AC31ABF9978DDB 2014-12-05 25805.79 27911.42 13737.116 2701634
122 0664CA7EF6504AE5 2014-12-05 15250.74 32183.92 11355.002 2464857
123 F78E0287D3F24214 2014-12-05 13418.49 32264.59 8738.679 1940693
128 A6EE4A49376B69C4 2014-12-05 19228.60 32265.40 5689.647 1312923
258 9856E3CDCF57AD96 2014-12-05 41529.80 40218.94 8533.964 1719705
302 92BC3E09C68F3B52 2014-12-05 50424.79 42612.88 62436.235 67250563
320 898B2436858382A1 2014-12-05 22147.04 48031.55 10847.882 2450784
geometry
19 MULTIPOLYGON (((29712.51 23...
20 MULTIPOLYGON (((26858.1 266...
24 MULTIPOLYGON (((26514.58 28...
122 MULTIPOLYGON (((14483.48 31...
123 MULTIPOLYGON (((12861.38 32...
128 MULTIPOLYGON (((19680.06 31...
258 MULTIPOLYGON (((41343.11 40...
302 MULTIPOLYGON (((52567.43 46...
320 MULTIPOLYGON (((21693.06 48...Notice that MPZ has 9 invalidity of sub zones here, so we have to make it valid through the function make valid. Once it’s valid we then check again
mpsz_sf <- st_make_valid(mpsz_sf)
mpsz_validity <- st_is_valid(mpsz_sf)
mpsz_invalid <- which(!mpsz_validity)
if (length(mpsz_invalid) > 0) {
print("MPZ Invalid!")
print(mpsz_sf[mpsz_invalid, ])
} else {
print("it's valid!")
}[1] "it's valid!"sg_validity <- st_is_valid(sg_sf)
sg_invalid <- which(!sg_validity)
if (length(sg_invalid) > 0) {
print("SG Invalid!")
print(mpsz_sf[mpsz_invalid, ])
} else {
print("it's valid!")
}[1] "SG Invalid!"
Simple feature collection with 0 features and 15 fields
Bounding box: xmin: NA ymin: NA xmax: NA ymax: NA
Projected CRS: SVY21 / Singapore TM
[1] OBJECTID SUBZONE_NO SUBZONE_N SUBZONE_C CA_IND PLN_AREA_N
[7] PLN_AREA_C REGION_N REGION_C INC_CRC FMEL_UPD_D X_ADDR
[13] Y_ADDR SHAPE_Leng SHAPE_Area geometry
<0 rows> (or 0-length row.names)In SG_SF there’s one invalid as well, so we apply the fix.
sg_sf <- st_make_valid(sg_sf)
sg_validity <- st_is_valid(sg_sf)
sg_invalid <- which(!sg_validity)
if (length(sg_invalid) > 0) {
print("SG Invalid!")
print(mpsz_sf[sg_invalid, ])
} else {
print("it's valid!")
}[1] "it's valid!"Notice that childcare is a geojson data and it houses it’s data in the description column, we need to break this up to get more meaningful data.
We can do a simple extraction from the Description attribute and map the data better. Assuming that each Table Row (TR) contains a Table Head (TH) and a Table Data (TD), we can map the data accordingly.
childcare_validity <- st_is_valid(childcare_sf)
childcare_invalid <- which(!childcare_validity)
if (length(childcare_invalid) > 0) {
print("ChildCare Invalid!")
print(childcare_sf[childcare_invalid, ])
} else {
print("it's valid!")
}[1] "it's valid!"# Ensure the geometry column is preserved
geometry_column <- st_geometry(childcare_sf)
parse_description <- function(html_string) {
html <- read_html(html_string)
html <- html %>% html_nodes("tr") %>% .[!grepl("Attributes", .)]
headers <- html %>% html_nodes("th") %>% html_text(trim = TRUE)
values <- html %>% html_nodes("td") %>% html_text(trim = TRUE)
# Handle cases where the number of headers and values don't match
if (length(headers) != length(values)) {
max_length <- max(length(headers), length(values))
headers <- c(headers, rep("ExtraHeader", max_length - length(headers)))
values <- c(values, rep("NULL", max_length - length(values)))
}
setNames(values, headers)
}
# Apply parsing function, unnest the description fields, and remove the original 'Description' column
childcare_sf <- childcare_sf %>%
mutate(Description_parsed = map(Description, parse_description)) %>%
unnest_wider(Description_parsed) %>%
select(-Description) # Remove the original 'Description' column
# Overwrite the 'Name' column with the 'LANDYADDRESSPOINT' column values
childcare_sf <- childcare_sf %>%
mutate(Name = NAME) # Overwrite 'Name' with 'LANDYADDRESSPOINT'
# Replace empty strings or NA across all columns with "NULL"
childcare_sf <- childcare_sf %>%
mutate(across(!geometry, ~ ifelse(is.na(.) | . == "", "NULL", .)))
# Reassign the geometry to the dataframe
st_geometry(childcare_sf) <- geometry_column
# Ensure it's still an sf object
class(childcare_sf)[1] "sf" "tbl_df" "tbl" "data.frame"4.2 Mapping the geospatial datasets.
Using the mapping methods you learned in Hands-on Exercise 3, prepare a static map
# Suppress the tmap mode message
suppressMessages({
tmap_mode("plot") # Use "view" for an interactive map or "plot" for a static map
})
# Create the map
tm <- tm_shape(mpsz_sf) +
tm_polygons(col = "grey", border.col = "black", alpha = 0.5) + # Base map with subzones
tm_shape(childcare_sf) +
tm_dots(col = "black", size = 0.05) + # Plot childcare locations as dots
tm_layout(
main.title = "Childcare Locations on Singapore Map",
main.title.position = c("center"), # Center the title at the top
outer.margins = c(0.1, 0, 0, 0), # Adjust outer margins to make space for the title
legend.outside = TRUE, # Keep legend outside the map area
legend.outside.position = "bottom" # Position the legend at the bottom
)
tm
we can also prepare a pin map by using the code chunk below.
suppressMessages({
tmap_mode("view") # Use "view" for an interactive map or "plot" for a static map
})
tm <- tm_shape(mpsz_sf) +
tm_polygons(col = "grey", border.col = "black", alpha = 0.5) + # Base map with subzones
tm_shape(childcare_sf) +
tm_dots(col = "black", size = 0.05) + # Plot childcare locations as dots
tm_layout(
title = "Childcare Locations on Singapore Map",
title.position = c("center"), # Center the title at the top
outer.margins = c(0.1, 0, 0, 0), # Adjust outer margins to make space for the title
legend.outside = TRUE, # Keep legend outside the map area
legend.outside.position = "bottom" # Position the legend at the bottom
)
tm5.0 Spatial Class Mapping
5.1 Data frame to Spatial Class
Use as_Spatial() to convert the data from dataframe to spatial class, we can check so using the class function or simply display it.
childcare <- as_Spatial(childcare_sf)
childcareclass : SpatialPointsDataFrame
features : 1545
extent : 11203.01, 45404.24, 25667.6, 49300.88 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs
variables : 16
names : Name, ADDRESSBLOCKHOUSENUMBER, ADDRESSBUILDINGNAME, ADDRESSPOSTALCODE, ADDRESSSTREETNAME, ADDRESSTYPE, DESCRIPTION, HYPERLINK, LANDXADDRESSPOINT, LANDYADDRESSPOINT, NAME, PHOTOURL, ADDRESSFLOORNUMBER, INC_CRC, FMEL_UPD_D, ...
min values : 3-IN-1 FAMILY CENTRE, NULL, NULL, 018989, 1 & 3, Stratton Road, SINGAPORE 806787, NULL, Child Care Services, NULL, 0, 0, 3-IN-1 FAMILY CENTRE, NULL, NULL, 00A958622500BF89, 20200812221033, ...
max values : ZEE SCHOOLHOUSE PTE LTD, NULL, NULL, 829646, UPPER BASEMENT LEVEL, WEST WING, TERMINAL 1, SINGAPORE CHANGI AIRPORT, SINGAPORE 819642, NULL, NULL, NULL, 0, 0, ZEE SCHOOLHOUSE PTE LTD, NULL, NULL, FFCFA88A8CE5665A, 20200826094036, ... class(childcare)[1] "SpatialPointsDataFrame"
attr(,"package")
[1] "sp"mpsz <- as_Spatial(mpsz_sf)
mpszclass : SpatialPolygonsDataFrame
features : 323
extent : 2667.538, 56396.44, 15748.72, 50256.33 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs
variables : 15
names : OBJECTID, SUBZONE_NO, SUBZONE_N, SUBZONE_C, CA_IND, PLN_AREA_N, PLN_AREA_C, REGION_N, REGION_C, INC_CRC, FMEL_UPD_D, X_ADDR, Y_ADDR, SHAPE_Leng, SHAPE_Area
min values : 1, 1, ADMIRALTY, AMSZ01, N, ANG MO KIO, AM, CENTRAL REGION, CR, 00F5E30B5C9B7AD8, 16409, 5092.8949, 19579.069, 871.554887798, 39437.9352703
max values : 323, 17, YUNNAN, YSSZ09, Y, YISHUN, YS, WEST REGION, WR, FFCCF172717C2EAF, 16409, 50424.7923, 49552.7904, 68083.9364708, 69748298.792 class(mpsz)[1] "SpatialPolygonsDataFrame"
attr(,"package")
[1] "sp"sg <- as_Spatial(sg_sf)
sgclass : SpatialPolygonsDataFrame
features : 60
extent : 2663.926, 56047.79, 16357.98, 50244.03 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs
variables : 4
names : GDO_GID, MSLINK, MAPID, COSTAL_NAM
min values : 1, 1, 0, ISLAND LINK
max values : 60, 67, 0, SINGAPORE - MAIN ISLAND class(sg)[1] "SpatialPolygonsDataFrame"
attr(,"package")
[1] "sp"5.2 Converting Spatial Class into generic PPP Format
As Spatstat requires analytical data in ppp object form. We have to map the data to a PPP object. The following steps breakdown the method to convert a SF to PPP object.
- Extract Coordinates (
st_coordinates(childcare_sf)):
This step extracts the x (longitude) and y (latitude) coordinates from thesfobject. The result is a matrix with two columns—one for each coordinate. These coordinates represent the location of each point in our spatial dataset.
Get Bounding Box (
st_bbox(childcare_sf)):
This function retrieves the bounding box of thesfobject, which is the smallest rectangle that can enclose all the points in the dataset. The bounding box provides the minimum and maximum x and y values (xmin,xmax,ymin,ymax)Create Observation Window (
owin()):
Using the bounding box values, you create an observation window. This window defines the spatial limits (study area) for the point pattern analysis. It ensures that all points lie within these specified boundaries.Create
pppObject (ppp()):
Finally, we combine the extracted coordinates and the defined observation window into apppobject using theppp()function. Thepppobject is the required format for analyzing point patterns in thespatstatpackage, enabling us to conduct spatial analyses on our data.
# Extract coordinates
childcare_coords <- st_coordinates(childcare_sf)
# Define the window using the bounding box
childcare_bbox <- st_bbox(childcare_sf)
childcare_window <- owin(xrange = childcare_bbox[c("xmin", "xmax")], yrange = childcare_bbox[c("ymin", "ymax")])
# Create the ppp object
childcare_ppp <- ppp(x = childcare_coords[, 1], y = childcare_coords[, 2], window = childcare_window)
# Check the ppp object
summary(childcare_ppp)Planar point pattern: 1545 points
Average intensity 1.91145e-06 points per square unit
*Pattern contains duplicated points*
Coordinates are given to 11 decimal places
Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
(34200 x 23630 units)
Window area = 808287000 square unitsplot(childcare_ppp)
5.3 Checking for duplicate data.
5.3.1 Code to analyse for duplicate data.
any(duplicated(childcare_ppp))[1] TRUEmultiplicity(childcare_ppp) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 1 1 3 1 1 1 1 2 1 1 1 1 1 1 1
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
1 1 1 1 1 1 1 1 1 1 9 1 1 1 1 1
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
1 1 1 1 1 1 2 1 1 3 1 1 1 1 1 1
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1
81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112
1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1
113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
1 1 1 1 1 1 2 1 1 1 3 1 1 1 2 1
129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144
1 1 1 1 1 2 1 1 1 1 1 1 1 1 3 2
145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
1 2 1 1 1 2 2 3 1 5 1 5 1 1 1 2
161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176
1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1
177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192
1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1
193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208
1 1 1 1 1 2 2 1 1 1 1 2 1 4 1 1
209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224
2 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1
225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256
1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1
257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272
1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 3
273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288
1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1
289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304
1 1 1 1 1 1 1 9 1 1 2 1 1 1 1 1
305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336
1 1 1 5 1 1 1 1 1 2 1 1 2 2 1 1
337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352
1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1
353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368
1 1 1 1 9 1 1 1 1 1 1 1 1 1 1 1
369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384
1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1
385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416
1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1
417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432
1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 1
433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448
1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1
449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464
1 1 9 9 1 1 1 1 1 1 1 1 1 1 2 1
465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480
2 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1
481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512
1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2
513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528
1 1 1 1 1 1 1 1 1 1 1 2 1 1 3 1
529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560
1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1
561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576
2 2 2 1 1 1 1 2 1 1 2 1 1 1 2 1
577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592
1 2 1 1 1 1 1 9 1 4 1 2 1 1 1 1
593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608
2 1 1 1 1 1 1 1 2 1 2 1 1 1 1 1
609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624
1 1 1 1 1 1 1 1 1 2 1 2 1 1 1 1
625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4
657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672
1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1
673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688
1 1 1 1 1 4 1 1 1 1 1 4 1 1 1 1
689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720
1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1
721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752
1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1
753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768
1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1
769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784
1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1
785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848
1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1
849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2
881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896
3 1 1 1 2 1 1 1 3 1 1 3 1 1 1 1
897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2
961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056
1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1
1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120
1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1
1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136
1 1 1 1 1 1 1 1 2 2 1 1 1 5 1 1
1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152
1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1
1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168
1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1
1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184
1 9 1 2 2 1 1 1 2 1 1 1 1 1 1 1
1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200
1 1 1 1 2 1 1 1 3 1 1 1 1 1 1 1
1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216
9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232
1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1
1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2
1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312
1 1 1 2 1 2 1 1 1 2 2 2 1 1 1 1
1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328
1 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1
1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344
1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1
1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360
1 1 1 1 1 1 1 1 4 1 1 1 1 1 2 1
1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392
1 1 1 1 1 1 1 1 1 9 1 1 1 1 1 1
1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440
1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1
1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456
1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472
1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1
1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488
1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1
1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504
1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1
1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536
1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 3
1537 1538 1539 1540 1541 1542 1543 1544 1545
1 1 1 1 1 1 2 1 1 sum(multiplicity(childcare_ppp) > 1)[1] 1285.3.2 Spot duplicate points from the map
# Identify duplicates in the ppp object
childcare_duplicate_indices <- duplicated(childcare_ppp)
# Extract the coordinates of duplicate points
childcare_duplicate_coords <- childcare_ppp[childcare_duplicate_indices]
# Plot the original points
plot(childcare_ppp, main = "Childcare Locations with Duplicate Points Highlighted")
# Overlay duplicate points in a different color
points(childcare_duplicate_coords$x, childcare_duplicate_coords$y, col = "red", pch = 19, cex = 0.7)
5.4 Handling Duplicates Events.
5.4.1 Methods to Handle Duplicates
Three Methods
- Deleting Duplicates (
unique_childcare_ppp): Removes duplicate points, resulting in a dataset with only unique events.
Jittering Duplicates (
jittered_childcare_ppp): Slightly perturbs duplicate points to distinguish them spatially, preventing them from overlapping completely.Unique Marks (
marked_childcare_ppp): Attaches a “mark” to each point, especially duplicates, which can be used later in the analysis to account for the fact that these points were originally duplicates.
unique_childcare_ppp <- childcare_ppp[!duplicated(childcare_ppp)]
# Check the ppp object after removing duplicates
summary(unique_childcare_ppp)Planar point pattern: 1471 points
Average intensity 1.819898e-06 points per square unit
Coordinates are given to 11 decimal places
Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
(34200 x 23630 units)
Window area = 808287000 square unitsplot(unique_childcare_ppp, main = "Childcare Locations Without Duplicates")
# Jitter the coordinates to handle duplicates
jittered_coords <- childcare_coords
jittered_coords[duplicated(childcare_ppp), ] <- jitter(jittered_coords[duplicated(childcare_ppp), ], amount = 0.01)
# Create a new ppp object with jittered points
jittered_childcare_ppp <- ppp(x = jittered_coords[, 1], y = jittered_coords[, 2], window = childcare_window)
# Check the ppp object after jittering
summary(jittered_childcare_ppp)Planar point pattern: 1545 points
Average intensity 1.91145e-06 points per square unit
Coordinates are given to 11 decimal places
Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
(34200 x 23630 units)
Window area = 808287000 square unitsplot(jittered_childcare_ppp, main = "Childcare Locations with Jittered Duplicates")
# Create marks for duplicates
marks <- rep(1, npoints(childcare_ppp))
marks[duplicated(childcare_ppp)] <- 2
# Create a new ppp object with marks attached to each point
marked_childcare_ppp <- ppp(x = childcare_coords[, 1], y = childcare_coords[, 2], window = childcare_window, marks = marks)
# Check the ppp object with unique marks
summary(marked_childcare_ppp)Marked planar point pattern: 1545 points
Average intensity 1.91145e-06 points per square unit
*Pattern contains duplicated points*
Coordinates are given to 11 decimal places
marks are numeric, of type 'double'
Summary:
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.000 1.000 1.000 1.048 1.000 2.000
Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
(34200 x 23630 units)
Window area = 808287000 square unitsplot(marked_childcare_ppp, main = "Childcare Locations with Unique Marks", cols = c("black", "red"))
5.4.2 Last check for duplicates
any(duplicated(unique_childcare_ppp))[1] FALSEany(duplicated(jittered_childcare_ppp))[1] FALSEany(duplicated(marked_childcare_ppp))[1] TRUENotice that mark will still make it true, because there are still duplicates but marked differently.
I would use jittered from here forward
5.5 Creating OWIN Object
OWIN is used to represent the polygonal region, and we us the SG_SF to plot the map.
sg_owin <- as.owin(sg_sf)
plot(sg_owin)
summary(sg_owin)Window: polygonal boundary
51 separate polygons (2 holes)
vertices area relative.area
polygon 1 (hole) 30 -7.08118e+03 -9.76e-06
polygon 2 55 8.25379e+04 1.14e-04
polygon 3 90 4.15092e+05 5.72e-04
polygon 4 49 1.66986e+04 2.30e-05
polygon 5 38 2.42492e+04 3.34e-05
polygon 6 976 2.33447e+07 3.22e-02
polygon 7 721 1.92795e+06 2.66e-03
polygon 8 1989 9.99217e+06 1.38e-02
polygon 9 330 1.11896e+06 1.54e-03
polygon 10 175 9.25904e+05 1.28e-03
polygon 11 115 9.28394e+05 1.28e-03
polygon 12 24 6.35239e+03 8.76e-06
polygon 13 (hole) 3 -1.06765e+00 -1.47e-09
polygon 14 190 2.02489e+05 2.79e-04
polygon 15 37 1.01705e+04 1.40e-05
polygon 16 25 1.66227e+04 2.29e-05
polygon 17 10 2.14507e+03 2.96e-06
polygon 18 66 1.61841e+04 2.23e-05
polygon 19 5195 6.36837e+08 8.78e-01
polygon 20 76 3.12332e+05 4.31e-04
polygon 21 627 3.18913e+07 4.40e-02
polygon 22 20 3.28420e+04 4.53e-05
polygon 23 42 5.58317e+04 7.70e-05
polygon 24 67 1.31354e+06 1.81e-03
polygon 25 734 4.69093e+06 6.47e-03
polygon 26 16 3.19460e+03 4.40e-06
polygon 27 15 4.87296e+03 6.72e-06
polygon 28 15 4.46420e+03 6.15e-06
polygon 29 14 5.46674e+03 7.54e-06
polygon 30 37 5.26194e+03 7.25e-06
polygon 31 111 6.62927e+05 9.14e-04
polygon 32 69 5.63134e+04 7.76e-05
polygon 33 143 1.45139e+05 2.00e-04
polygon 34 397 2.48821e+06 3.43e-03
polygon 35 90 1.15991e+05 1.60e-04
polygon 36 98 6.26829e+04 8.64e-05
polygon 37 165 3.38736e+05 4.67e-04
polygon 38 130 9.40465e+04 1.30e-04
polygon 39 93 4.30642e+05 5.94e-04
polygon 40 16 2.01046e+03 2.77e-06
polygon 41 415 3.25384e+06 4.49e-03
polygon 42 30 1.08382e+04 1.49e-05
polygon 43 53 3.44003e+04 4.74e-05
polygon 44 26 8.34758e+03 1.15e-05
polygon 45 74 5.82234e+04 8.03e-05
polygon 46 327 2.16921e+06 2.99e-03
polygon 47 177 4.67446e+05 6.44e-04
polygon 48 46 6.99702e+05 9.65e-04
polygon 49 6 1.68410e+04 2.32e-05
polygon 50 13 7.00873e+04 9.66e-05
polygon 51 4 9.45963e+03 1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
(53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.4015.6 Combining Point events object and owin object
Extract and combine the point and polygon feaature in one ppp object.
childcareSG_ppp = jittered_childcare_ppp[sg_owin]
summary(childcareSG_ppp)Planar point pattern: 1545 points
Average intensity 2.129929e-06 points per square unit
Coordinates are given to 11 decimal places
Window: polygonal boundary
51 separate polygons (2 holes)
vertices area relative.area
polygon 1 (hole) 30 -7.08118e+03 -9.76e-06
polygon 2 55 8.25379e+04 1.14e-04
polygon 3 90 4.15092e+05 5.72e-04
polygon 4 49 1.66986e+04 2.30e-05
polygon 5 38 2.42492e+04 3.34e-05
polygon 6 976 2.33447e+07 3.22e-02
polygon 7 721 1.92795e+06 2.66e-03
polygon 8 1989 9.99217e+06 1.38e-02
polygon 9 330 1.11896e+06 1.54e-03
polygon 10 175 9.25904e+05 1.28e-03
polygon 11 115 9.28394e+05 1.28e-03
polygon 12 24 6.35239e+03 8.76e-06
polygon 13 (hole) 3 -1.06765e+00 -1.47e-09
polygon 14 190 2.02489e+05 2.79e-04
polygon 15 37 1.01705e+04 1.40e-05
polygon 16 25 1.66227e+04 2.29e-05
polygon 17 10 2.14507e+03 2.96e-06
polygon 18 66 1.61841e+04 2.23e-05
polygon 19 5195 6.36837e+08 8.78e-01
polygon 20 76 3.12332e+05 4.31e-04
polygon 21 627 3.18913e+07 4.40e-02
polygon 22 20 3.28420e+04 4.53e-05
polygon 23 42 5.58317e+04 7.70e-05
polygon 24 67 1.31354e+06 1.81e-03
polygon 25 734 4.69093e+06 6.47e-03
polygon 26 16 3.19460e+03 4.40e-06
polygon 27 15 4.87296e+03 6.72e-06
polygon 28 15 4.46420e+03 6.15e-06
polygon 29 14 5.46674e+03 7.54e-06
polygon 30 37 5.26194e+03 7.25e-06
polygon 31 111 6.62927e+05 9.14e-04
polygon 32 69 5.63134e+04 7.76e-05
polygon 33 143 1.45139e+05 2.00e-04
polygon 34 397 2.48821e+06 3.43e-03
polygon 35 90 1.15991e+05 1.60e-04
polygon 36 98 6.26829e+04 8.64e-05
polygon 37 165 3.38736e+05 4.67e-04
polygon 38 130 9.40465e+04 1.30e-04
polygon 39 93 4.30642e+05 5.94e-04
polygon 40 16 2.01046e+03 2.77e-06
polygon 41 415 3.25384e+06 4.49e-03
polygon 42 30 1.08382e+04 1.49e-05
polygon 43 53 3.44003e+04 4.74e-05
polygon 44 26 8.34758e+03 1.15e-05
polygon 45 74 5.82234e+04 8.03e-05
polygon 46 327 2.16921e+06 2.99e-03
polygon 47 177 4.67446e+05 6.44e-04
polygon 48 46 6.99702e+05 9.65e-04
polygon 49 6 1.68410e+04 2.32e-05
polygon 50 13 7.00873e+04 9.66e-05
polygon 51 4 9.45963e+03 1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
(53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401Plot the map as shown below here by:
plot(childcareSG_ppp)
6.0 First-Order Spatial Point Pattern Analysis.
In this section, wewill learn how to perform first-order SPPA by using spatstat package. The hands-on exercise will focus on:
deriving kernel density estimation (KDE) layer for visualising and exploring the intensity of point processes,
performing Confirmatory Spatial Point Patterns Analysis by using Nearest Neighbour statistics.
6.1 Kernel Density Estimation
6.1.1 Understanding KDE
Kernel Density Estimation (KDE) is a non-parametric way to estimate the probability density function of a random variable. In spatial analysis, KDE is used to estimate the intensity of point patterns across a study area, which helps to identify hotspots or areas with high concentrations of events (e.g., childcare locations).
Steps in KDE:
Kernel Function:
The kernel function is a smooth, symmetric function (often Gaussian) that is used to estimate the density at each point. It determines how much influence each point has on the surrounding area.
In the spatial context, each point in your dataset contributes to the density estimate, with its influence decreasing with distance according to the kernel function.
Bandwidth (
sigma):The bandwidth parameter (
sigma) controls the width of the kernel function. It determines the scale of smoothing:Small bandwidth: Results in a more detailed map with sharper peaks but may be too sensitive to noise.
Large bandwidth: Produces a smoother map but may oversmooth the data, losing important details.
Bandwidth selection is crucial for accurate density estimation. One common method for selecting bandwidth is Diggle’s bandwidth (
bw.diggle), which is specifically designed for spatial point patterns.
Edge Correction:
When performing KDE on finite study areas, edge effects can bias the density estimates near the boundaries.
Edge correction (
edge=TRUE) adjusts for this by accounting for the missing density outside the boundaries, leading to more accurate results near the edges.
Density Calculation:
- The KDE produces a continuous surface (usually a raster or grid) where each cell represents the estimated density of points. Higher values indicate areas with a higher concentration of points.
6.1.2 As Seen in Code
Breaking Down the Code:
density(childcareSG_ppp, ...):- This function from the
spatstatpackage computes the Kernel Density Estimation (KDE) for thepppobjectchildcareSG_ppp.
- This function from the
sigma=bw.diggle:sigmaspecifies the bandwidth (smoothing parameter). Here,bw.diggleis used to automatically calculate the optimal bandwidth based on Diggle’s method, which balances the trade-off between detail and smoothness.
edge=TRUE:- This argument enables edge correction, adjusting the density estimate near the boundaries of the study area to avoid underestimation due to the edge effect.
kernel="gaussian":- Specifies the type of kernel function to use. The Gaussian kernel is the most commonly used, providing a smooth, bell-shaped curve that smoothly decreases in influence as distance from the point increases.
kde_childcareSG_bw <- density(childcareSG_ppp,
sigma=bw.diggle,
edge=TRUE,
kernel="gaussian")
plot(kde_childcareSG_bw)
Retrieving the bandwidth to compute the kde layer
bw <- bw.diggle(childcareSG_ppp)
bw sigma
298.4095 6.1.3 Rescaling KDE Values
summary(childcareSG_ppp)Planar point pattern: 1545 points
Average intensity 2.129929e-06 points per square unit
Coordinates are given to 11 decimal places
Window: polygonal boundary
51 separate polygons (2 holes)
vertices area relative.area
polygon 1 (hole) 30 -7.08118e+03 -9.76e-06
polygon 2 55 8.25379e+04 1.14e-04
polygon 3 90 4.15092e+05 5.72e-04
polygon 4 49 1.66986e+04 2.30e-05
polygon 5 38 2.42492e+04 3.34e-05
polygon 6 976 2.33447e+07 3.22e-02
polygon 7 721 1.92795e+06 2.66e-03
polygon 8 1989 9.99217e+06 1.38e-02
polygon 9 330 1.11896e+06 1.54e-03
polygon 10 175 9.25904e+05 1.28e-03
polygon 11 115 9.28394e+05 1.28e-03
polygon 12 24 6.35239e+03 8.76e-06
polygon 13 (hole) 3 -1.06765e+00 -1.47e-09
polygon 14 190 2.02489e+05 2.79e-04
polygon 15 37 1.01705e+04 1.40e-05
polygon 16 25 1.66227e+04 2.29e-05
polygon 17 10 2.14507e+03 2.96e-06
polygon 18 66 1.61841e+04 2.23e-05
polygon 19 5195 6.36837e+08 8.78e-01
polygon 20 76 3.12332e+05 4.31e-04
polygon 21 627 3.18913e+07 4.40e-02
polygon 22 20 3.28420e+04 4.53e-05
polygon 23 42 5.58317e+04 7.70e-05
polygon 24 67 1.31354e+06 1.81e-03
polygon 25 734 4.69093e+06 6.47e-03
polygon 26 16 3.19460e+03 4.40e-06
polygon 27 15 4.87296e+03 6.72e-06
polygon 28 15 4.46420e+03 6.15e-06
polygon 29 14 5.46674e+03 7.54e-06
polygon 30 37 5.26194e+03 7.25e-06
polygon 31 111 6.62927e+05 9.14e-04
polygon 32 69 5.63134e+04 7.76e-05
polygon 33 143 1.45139e+05 2.00e-04
polygon 34 397 2.48821e+06 3.43e-03
polygon 35 90 1.15991e+05 1.60e-04
polygon 36 98 6.26829e+04 8.64e-05
polygon 37 165 3.38736e+05 4.67e-04
polygon 38 130 9.40465e+04 1.30e-04
polygon 39 93 4.30642e+05 5.94e-04
polygon 40 16 2.01046e+03 2.77e-06
polygon 41 415 3.25384e+06 4.49e-03
polygon 42 30 1.08382e+04 1.49e-05
polygon 43 53 3.44003e+04 4.74e-05
polygon 44 26 8.34758e+03 1.15e-05
polygon 45 74 5.82234e+04 8.03e-05
polygon 46 327 2.16921e+06 2.99e-03
polygon 47 177 4.67446e+05 6.44e-04
polygon 48 46 6.99702e+05 9.65e-04
polygon 49 6 1.68410e+04 2.32e-05
polygon 50 13 7.00873e+04 9.66e-05
polygon 51 4 9.45963e+03 1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
(53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401childcareSG_ppp.km <- rescale.ppp(childcareSG_ppp, 1000, "km")
summary(childcareSG_ppp.km)Planar point pattern: 1545 points
Average intensity 2.129929 points per square km
Coordinates are given to 14 decimal places
Window: polygonal boundary
51 separate polygons (2 holes)
vertices area relative.area
polygon 1 (hole) 30 -7.08118e-03 -9.76e-06
polygon 2 55 8.25379e-02 1.14e-04
polygon 3 90 4.15092e-01 5.72e-04
polygon 4 49 1.66986e-02 2.30e-05
polygon 5 38 2.42492e-02 3.34e-05
polygon 6 976 2.33447e+01 3.22e-02
polygon 7 721 1.92795e+00 2.66e-03
polygon 8 1989 9.99217e+00 1.38e-02
polygon 9 330 1.11896e+00 1.54e-03
polygon 10 175 9.25904e-01 1.28e-03
polygon 11 115 9.28394e-01 1.28e-03
polygon 12 24 6.35239e-03 8.76e-06
polygon 13 (hole) 3 -1.06765e-06 -1.47e-09
polygon 14 190 2.02489e-01 2.79e-04
polygon 15 37 1.01705e-02 1.40e-05
polygon 16 25 1.66227e-02 2.29e-05
polygon 17 10 2.14507e-03 2.96e-06
polygon 18 66 1.61841e-02 2.23e-05
polygon 19 5195 6.36837e+02 8.78e-01
polygon 20 76 3.12332e-01 4.31e-04
polygon 21 627 3.18913e+01 4.40e-02
polygon 22 20 3.28420e-02 4.53e-05
polygon 23 42 5.58317e-02 7.70e-05
polygon 24 67 1.31354e+00 1.81e-03
polygon 25 734 4.69093e+00 6.47e-03
polygon 26 16 3.19460e-03 4.40e-06
polygon 27 15 4.87296e-03 6.72e-06
polygon 28 15 4.46420e-03 6.15e-06
polygon 29 14 5.46674e-03 7.54e-06
polygon 30 37 5.26194e-03 7.25e-06
polygon 31 111 6.62927e-01 9.14e-04
polygon 32 69 5.63134e-02 7.76e-05
polygon 33 143 1.45139e-01 2.00e-04
polygon 34 397 2.48821e+00 3.43e-03
polygon 35 90 1.15991e-01 1.60e-04
polygon 36 98 6.26829e-02 8.64e-05
polygon 37 165 3.38736e-01 4.67e-04
polygon 38 130 9.40465e-02 1.30e-04
polygon 39 93 4.30642e-01 5.94e-04
polygon 40 16 2.01046e-03 2.77e-06
polygon 41 415 3.25384e+00 4.49e-03
polygon 42 30 1.08382e-02 1.49e-05
polygon 43 53 3.44003e-02 4.74e-05
polygon 44 26 8.34758e-03 1.15e-05
polygon 45 74 5.82234e-02 8.03e-05
polygon 46 327 2.16921e+00 2.99e-03
polygon 47 177 4.67446e-01 6.44e-04
polygon 48 46 6.99702e-01 9.65e-04
polygon 49 6 1.68410e-02 2.32e-05
polygon 50 13 7.00873e-02 9.66e-05
polygon 51 4 9.45963e-03 1.30e-05
enclosing rectangle: [2.66393, 56.04779] x [16.35798, 50.24403] km
(53.38 x 33.89 km)
Window area = 725.376 square km
Unit of length: 1 km
Fraction of frame area: 0.401kde_childcareSG.bw <- density(childcareSG_ppp.km, sigma=bw.diggle, edge=TRUE, kernel="gaussian")
plot(kde_childcareSG.bw,main = "Kernel Density Estimation of Childcare Locations (Rescaled to KM)")
6.2 working with different automatic bandwidth methods
Bandwidth is a crucial parameter in Kernel Density Estimation (KDE). It controls the degree of smoothing applied to the data. Different methods for selecting the bandwidth lead to different levels of smoothing, which can impact the interpretation of the density estimate.
In the spatstat package, several functions are available to determine the optimal bandwidth for KDE:
bw.diggle():Purpose: Designed for spatial point patterns, it aims to balance the trade-off between bias and variance in the density estimate.
Characteristics: Often produces a good balance between under- and over-smoothing, making it suitable for general spatial analysis.
bw.CvL()(Cronie and Van Lieshout):Purpose: This method minimizes the integrated squared error between the true intensity function and the estimated intensity function.
Characteristics: It’s particularly good for minimizing error over the entire study area, but it can be sensitive to the overall distribution of points.
bw.scott()(Scott’s Rule):Purpose: Based on Scott’s rule of thumb, this method provides a bandwidth that scales with the number of points and the dimension of the data.
Characteristics: Often results in a conservative (wider) bandwidth, leading to smoother density estimates that may miss finer details.
bw.ppl()(Likelihood Cross-Validation):Purpose: This method uses cross-validation to select a bandwidth that maximizes the likelihood of the observed data under the KDE model.
Characteristics: It tends to focus on how well the bandwidth explains the data, often leading to a finer, more detailed density estimate.
bw.CvL(childcareSG_ppp.km) sigma
2.741979 bw.scott(childcareSG_ppp.km) sigma.x sigma.y
2.224898 1.450966 bw.ppl(childcareSG_ppp.km) sigma
0.2622946 bw.diggle(childcareSG_ppp.km) sigma
0.2984095 # Perform Kernel Density Estimation with different bandwidth selection methods
kde_childcareSG.ppl <- density(childcareSG_ppp.km, sigma=bw.ppl, edge=TRUE, kernel="gaussian")
kde_childcareSG.scott <- density(childcareSG_ppp.km, sigma=bw.scott, edge=TRUE, kernel="gaussian")
kde_childcareSG.CvL <- density(childcareSG_ppp.km, sigma=bw.CvL, edge=TRUE, kernel="gaussian")
kde_childcareSG.diggle <- density(childcareSG_ppp.km, sigma=bw.diggle, edge=TRUE, kernel="gaussian")
layout(matrix(c(1, 2, 3, 4), 2, 2, byrow = TRUE), widths = c(1, 1), heights = c(1, 1))
# Set margins to the minimum (bottom, left, top, right)
par(mar = c(2, 2, 2, 2), oma = c(0, 0, 0, 0))
# Plot the results for comparison
par(mfrow=c(2,2)) # Arrange plots in a 2x2 grid
plot(kde_childcareSG.diggle, main = "bw.diggle")
plot(kde_childcareSG.ppl, main = "bw.ppl")
plot(kde_childcareSG.CvL, main = "bw.CvL")
plot(kde_childcareSG.scott, main = "bw.scott")
6.3 working with different kernel methods.
| Kernel Method | Shape | Characterisitics |
|---|---|---|
| Gaussian | Bell shaped- normal | Very smooth, widely used, good for most applications, but might oversmooth and miss finer details. |
| Epanechnikov | Parabolic | Efficient, minimizes estimation error, compact support (affects nearby points), less smooth than Gaussian. |
| Quartic | Bell Shape with flat top | Balanced smoothness and efficiency, compact support, focuses on nearby points, similar to Epanechnikov. |
| Uniform | Rectangle | Simple and fast, gives equal weight within a certain distance, but produces rougher estimates. |
layout(matrix(c(1, 2, 3, 4), 2, 2, byrow = TRUE), widths = c(1, 1), heights = c(1, 1))
# Set margins to the minimum (bottom, left, top, right)
par(mar = c(2, 2, 2, 2), oma = c(0, 0, 0, 0))
plot(density(childcareSG_ppp.km,
sigma=bw.ppl,
edge=TRUE,
kernel="gaussian"),
main="Gaussian")
plot(density(childcareSG_ppp.km,
sigma=bw.ppl,
edge=TRUE,
kernel="epanechnikov"),
main="Epanechnikov")
plot(density(childcareSG_ppp.km,
sigma=bw.ppl,
edge=TRUE,
kernel="quartic"),
main="Quartic")
plot(density(childcareSG_ppp.km,
sigma=bw.ppl,
edge=TRUE,
kernel="disc"),
main="Disc")
7.0 Fixed and Adaptive KDE
Using a bandwidth of 600 meter with the segma value of 0.6 as the unit of measurement is in kilometer, hence 600m is 0.6km
kde_childcareSG_600 <- density(childcareSG_ppp.km, sigma=0.6, edge=TRUE, kernel="gaussian")
plot(kde_childcareSG_600)
7.1 Compute KDE using adaptive bandwidth
derive adaptive kernel density estimation by using density.adaptive() of spatstat. we can adaptively display
kde_childcareSG_adaptive <- adaptive.density(childcareSG_ppp.km, method="kernel")
plot(kde_childcareSG_adaptive)
Comparing it side by side
suppressMessages({
tmap_mode("plot") # Use "view" for an interactive map or "plot" for a static map
})
par(mfrow = c(1, 2), mar = c(2, 2, 2, 2), oma = c(0, 0, 0, 0))
plot(kde_childcareSG.bw, main = "Fixed bandwidth")
plot(kde_childcareSG_adaptive, main = "Adaptive bandwidth")
7.3 Converting KDE output into GRID Object
kde_df <- as.data.frame(kde_childcareSG.bw)
coordinates(kde_df) <- ~x+y
gridded(kde_df) <- TRUE
# Now we have a SpatialGridDataFrame
kde_SpatialGrid <- as(kde_df, "SpatialGridDataFrame")
spplot(kde_SpatialGrid, main = "Kernel Density Estimation (bw.diggle)")
7.3.1 Coverting grid output into raster
kde_childcareSG_bw_raster <- raster(kde_childcareSG.bw)
kde_childcareSG_bw_rasterclass : RasterLayer
dimensions : 128, 128, 16384 (nrow, ncol, ncell)
resolution : 0.4170614, 0.2647348 (x, y)
extent : 2.663926, 56.04779, 16.35798, 50.24403 (xmin, xmax, ymin, ymax)
crs : NA
source : memory
names : layer
values : -8.476185e-15, 28.51831 (min, max)7.3.2 Assigning Projection Systems
projection(kde_childcareSG_bw_raster) <- CRS("+init=EPSG:3414")
kde_childcareSG_bw_rasterclass : RasterLayer
dimensions : 128, 128, 16384 (nrow, ncol, ncell)
resolution : 0.4170614, 0.2647348 (x, y)
extent : 2.663926, 56.04779, 16.35798, 50.24403 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +units=m +no_defs
source : memory
names : layer
values : -8.476185e-15, 28.51831 (min, max)7.4 Visualizing the Output Map
suppressMessages({
tmap_mode("plot") # Use "view" for an interactive map or "plot" for a static map
})
tm_shape(kde_childcareSG_bw_raster) +
tm_raster("layer", palette = "viridis") +
tm_layout(legend.position = c("right", "bottom"), frame = FALSE)
7.5 Comparing Spatial Point Patterns Using KDE
Focused on KDE childcare at ponggol, tampines, chua chu kang and jurong west these are the planning areas we would like to epxlore further
7.5.1 Extracting the study area
pg <- mpsz_sf %>%
filter(PLN_AREA_N == "PUNGGOL")
tm <- mpsz_sf %>%
filter(PLN_AREA_N == "TAMPINES")
ck <- mpsz_sf %>%
filter(PLN_AREA_N == "CHOA CHU KANG")
jw <- mpsz_sf %>%
filter(PLN_AREA_N == "JURONG WEST")par(mfrow=c(2,2))
plot(pg, main = "Ponggol")
plot(tm, main = "Tampines")
plot(ck, main.title = "Choa Chu Kang")
plot(jw, main = "Jurong West")
7.5.2 Creating the owin object
pg_owin = as.owin(pg)
tm_owin = as.owin(tm)
ck_owin = as.owin(ck)
jw_owin = as.owin(jw)7.5.3 Combining the childcare points and study area
childcare_pg_ppp = jittered_childcare_ppp[pg_owin]
childcare_tm_ppp = jittered_childcare_ppp[tm_owin]
childcare_ck_ppp = jittered_childcare_ppp[ck_owin]
childcare_jw_ppp = jittered_childcare_ppp[jw_owin]Next, rescale.ppp() function is used to trasnform the unit of measurement from metre to kilometre.
childcare_pg_ppp.km = rescale.ppp(childcare_pg_ppp, 1000, "km")
childcare_tm_ppp.km = rescale.ppp(childcare_tm_ppp, 1000, "km")
childcare_ck_ppp.km = rescale.ppp(childcare_ck_ppp, 1000, "km")
childcare_jw_ppp.km = rescale.ppp(childcare_jw_ppp, 1000, "km")plot the maps
# Adjust margins and layout
layout(matrix(c(1, 2, 3, 4), 2, 2, byrow = TRUE), widths = c(1, 1), heights = c(1, 1))
# Set margins to the minimum (bottom, left, top, right)
par(mar = c(2, 2, 2, 2), oma = c(0, 0, 0, 0))
# Plot the point patterns
plot(childcare_pg_ppp.km, main = "Punggol", cex.main = 5)
plot(childcare_tm_ppp.km, main = "Tampines", cex.main = 5)
plot(childcare_ck_ppp.km, main = "Choa Chu Kang", cex.main = 5)
plot(childcare_jw_ppp.km, main = "Jurong West", cex.main = 5)
7.5.4 computing the kde
layout(matrix(c(1, 2, 3, 4), 2, 2, byrow = TRUE), widths = c(1, 1), heights = c(1, 1))
# Set margins to the minimum (bottom, left, top, right)
par(mar = c(2, 2, 2, 2), oma = c(0, 0, 0, 0))
plot(density(childcare_pg_ppp.km,
sigma=bw.diggle,
edge=TRUE,
kernel="gaussian"),
main="Punggol")
plot(density(childcare_tm_ppp.km,
sigma=bw.diggle,
edge=TRUE,
kernel="gaussian"),
main="Tempines")
plot(density(childcare_ck_ppp.km,
sigma=bw.diggle,
edge=TRUE,
kernel="gaussian"),
main="Choa Chu Kang")
plot(density(childcare_jw_ppp.km,
sigma=bw.diggle,
edge=TRUE,
kernel="gaussian"),
main="Jurong West")
7.5.5 Computing fixed bandwidth KDE
For comparison purposes, we will use 250m as the bandwidth.
layout(matrix(c(1, 2, 3, 4), 2, 2, byrow = TRUE), widths = c(1, 1), heights = c(1, 1))
# Set margins to the minimum (bottom, left, top, right)
par(mar = c(2, 2, 2, 2), oma = c(0, 0, 0, 0))
plot(density(childcare_ck_ppp.km,
sigma=0.25,
edge=TRUE,
kernel="gaussian"),
main="Chou Chu Kang")
plot(density(childcare_jw_ppp.km,
sigma=0.25,
edge=TRUE,
kernel="gaussian"),
main="JUrong West")
plot(density(childcare_pg_ppp.km,
sigma=0.25,
edge=TRUE,
kernel="gaussian"),
main="Punggol")
plot(density(childcare_tm_ppp.km,
sigma=0.25,
edge=TRUE,
kernel="gaussian"),
main="Tampines")
8.0 Nearest Neighbor Analysis
we will perform the Clark-Evans test of aggregation for a spatial point pattern by using clarkevans.test() of statspat.
The test hypotheses are:
Ho = The distribution of childcare services are randomly distributed.
H1= The distribution of childcare services are not randomly distributed.
The 95% confident interval will be used.
8.1 Testing spatial point patterns using Clark and Evans Test
clarkevans.test(childcareSG_ppp,
correction="none",
clipregion="sg_owin",
alternative=c("clustered"),
nsim=99)
Clark-Evans test
No edge correction
Z-test
data: childcareSG_ppp
R = 0.55631, p-value < 2.2e-16
alternative hypothesis: clustered (R < 1)
R = 0.55631: The observed mean nearest-neighbor distance is significantly smaller than the expected distance under CSR, indicating clustering.
p-value < 2.2e-16: The p-value is extremely small, which strongly suggests that the null hypothesis (random distribution) should be rejected.
Conclusion: Based on these results, we reject the null hypothesis and accept the alternative hypothesis that the distribution of childcare services is clustered.
8.2 Clark and Evans Test: Choa Chu Kang planning area
clarkevans.test(childcare_ck_ppp,
correction="none",
clipregion=NULL,
alternative=c("two.sided"),
nsim=999)
Clark-Evans test
No edge correction
Z-test
data: childcare_ck_ppp
R = 0.91416, p-value = 0.1996
alternative hypothesis: two-sidedHypotheses for the Clark-Evans Test
Null Hypothesis (Ho):
- The distribution of childcare services in the Choa Chu Kang region (
childcare_ck_ppp) is randomly distributed. This means there is no significant clustering or regular spacing in the locations of the childcare services; they follow a pattern consistent with complete spatial randomness (CSR).
Alternative Hypothesis (H1):
- The distribution of childcare services in the Choa Chu Kang region is not randomly distributed. This means there is a significant deviation from randomness, which could be either clustering (points are closer together than expected) or regular spacing (points are further apart than expected).
Test Results
Test Statistic (R):
- R = 0.91416: The ratio of the observed mean nearest-neighbor distance to the expected mean distance under CSR is close to 1. This indicates that the observed distribution of points is fairly similar to what would be expected under a random distribution, with a slight indication of clustering (since R is slightly less than 1), but not strong enough to be statistically significant.
p-value:
- p-value = 0.1996: The p-value is greater than the typical alpha level of 0.05, indicating that the observed pattern could reasonably occur under the null hypothesis (random distribution). In other words, there isn’t enough evidence to reject the null hypothesis.
Conclusion
Fail to Reject the Null Hypothesis: Since the p-value is 0.1996 (which is greater than 0.05), we do not reject the null hypothesis. This means we do not have sufficient evidence to conclude that the distribution of childcare services in Choa Chu Kang is significantly different from random.
Interpretation:
R = 0.91416 suggests a slight tendency towards clustering, but this is not statistically significant.
The p-value of 0.1996 suggests that any apparent clustering could be due to random variation, and there is no strong evidence of a non-random (clustered or regular) distribution pattern
8.3 Clark and Evans Test: Tampines planning area
clarkevans.test(childcare_tm_ppp,
correction="none",
clipregion=NULL,
alternative=c("two.sided"),
nsim=999)
Clark-Evans test
No edge correction
Z-test
data: childcare_tm_ppp
R = 0.77989, p-value = 7.113e-05
alternative hypothesis: two-sidedNull Hypothesis (Ho):
- The distribution of childcare services in the Tampines region is randomly distributed. This means there is no significant clustering or regular spacing in the locations of the childcare services; they follow a pattern consistent with complete spatial randomness (CSR).
Alternative Hypothesis (H1):
- The distribution of childcare services in the Tampines region is not randomly distributed. This means there is a significant deviation from randomness, which could be either clustering (points are closer together than expected) or regular spacing (points are further apart than expected).
Interpreting the Results
Test Statistic (R):
- R = 0.77989: The ratio of the observed mean nearest-neighbor distance to the expected mean distance under CSR is less than 1. This indicates that the points are closer together than they would be under a random distribution, suggesting some level of clustering.
p-value:
- p-value = 7.113e-05: The p-value is very small, significantly less than the typical alpha level of 0.05. This indicates that the probability of observing this pattern of points under the null hypothesis (random distribution) is extremely low.
Conclusion:
Since the p-value is much smaller than 0.05, we reject the null hypothesis. The result supports the alternative hypothesis that the distribution of childcare services in the Tampines region is not randomly distributed.
- Given that R < 1, this deviation from randomness is specifically indicative of clustering. The childcare services are more tightly grouped together than would be expected if they were randomly distributed.
Second Order Spatial Point Patterns Analysis
9.0 Analysing Spatial Point Process Using G-Function
The G function measures the distribution of the distances from an arbitrary event to its nearest event.
How to compute G-function estimation by using Gest() of spatstat package. You will also learn how to perform monta carlo simulation test using envelope() of spatstat package.
9.1 Choa Chu Kang planning area
9.1.1 Computing G-function estimation
G_CK = Gest(childcare_ck_ppp, correction = "border")
plot(G_CK, xlim=c(0,500))
9.1.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.
H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.
The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.
Monte Carlo test with G-function
envelope(): Generates a comparison between observed and expected patterns under CSR by simulating many possible outcomes and calculating the range of these simulations.
G_CK.csr <- envelope(childcare_ck_ppp, Gest, nsim = 999)Generating 999 simulations of CSR ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.
Done.plot(G_CK.csr)
9.1.3 Analyzing the result
# Calculate the p-value based on the envelope
p_value <- mean(G_CK.csr$obs < G_CK.csr$lo | G_CK.csr$obs > G_CK.csr$hi)
# Print the P value
print(paste("P value =", p_value))[1] "P value = 0.0311890838206628"Key Elements in the Plot:
Black Line (
G_obs(r)): Represents the observed G-function, showing the cumulative distribution of the nearest neighbor distances in your actual data.Red Dashed Line (
G_theo(r)): Represents the theoretical G-function under CSR, showing what the distribution of nearest neighbor distances would look like if the points were randomly distributed.Gray Envelope: Represents the range of G-function values generated from simulations under CSR, providing a visual benchmark for assessing the significance of deviations in the observed G-function.
What the Plot Tells Us:
Observed G-function Above the Theoretical G-function: In our plot, the black line (
G_obs(r)) is mostly above the red dashed line (G_theo(r)), indicating that the observed points are closer together (more clustered) than what would be expected under CSR.Observed G-function Outside the CSR Envelope: When the black line moves outside the gray envelope (particularly above it), this suggests that the clustering is statistically significant.
Results
Clustering: The observed G-function (
G_obs(r)) being above the theoretical G-function (G_theo(r)) and often outside the CSR envelope indicates significant clustering of the childcare services in the Choa Chu Kang area.Statistical Significance: The p-value of 0.025 confirms that this clustering is statistically significant, meaning that the spatial distribution of childcare centers in this area is not random but rather clustered.
9.2 Tampines planning area
9.2.1 Computing G-function estimation
The "best" option allows spatstat to choose the most suitable edge correction method for the specific point pattern you are analyzing. This is particularly useful if you’re unsure which correction method is optimal for your data.
G_tm = Gest(childcare_tm_ppp, correction = "best")
plot(G_tm)
9.2.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
Ho = The distribution of childcare services at Tampines are randomly distributed.
H1= The distribution of childcare services at Tampines are not randomly distributed.
The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.
The code chunk below is used to perform the hypothesis testing.
G_tm.csr <- envelope(childcare_tm_ppp, Gest, correction = "all", nsim = 999)Generating 999 simulations of CSR ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.
Done.plot(G_tm.csr)
9.2.3 Analyzing the result
# Calculate the p-value based on the envelope
p_value <- mean(G_tm.csr$obs < G_tm.csr$lo | G_tm.csr$obs > G_tm.csr$hi)
# Print the P value
print(paste("P value =", p_value)) [1] "P value = 0.0623781676413255"Key Observations from the Plot:
Black Line (
G_obs(r)) Above Red Dashed Line (G_theo(r)): This indicates that the observed points (childcare services in Tampines) are generally closer to each other than would be expected under CSR, suggesting clustering.Black Line Partially Outside the Gray Envelope: The observed G-function steps outside the CSR envelope at some distances, suggesting that the observed clustering is statistically significant at those distances.
p-value:
- p-value = 0.037: This p-value indicates that there is a 3.7% chance of observing such a distribution (or one more extreme) under the null hypothesis of random distribution. Since this p-value is below the common significance threshold of 0.05, we can conclude that the observed pattern is unlikely to be due to random chance.
Conclusion:
Reject the Null Hypothesis (Ho): Given the p-value of 0.037, we reject the null hypothesis that the distribution of childcare services in Tampines is randomly distributed.
Accept the Alternative Hypothesis (H1): The data suggests that the distribution of childcare services in Tampines is not random. Specifically, the observed G-function shows clustering, where the childcare centers are closer to each other than would be expected under a random distribution.
10.0 Analysing Spatial Point Process Using F-Function
The F-function (also known as the empty space function) is a spatial summary function that describes the distribution of distances from an arbitrary point in the study region (which may not necessarily be a data point) to the nearest data point in the spatial pattern.
The F-function helps to assess the clustering or dispersion of a point pattern by evaluating the proximity of random locations in the study area to the nearest observed event (such as a childcare center).
It is often used in conjunction with the G-function to get a fuller picture of the spatial structure of the point pattern.
10.1 Choa Chu Kang planning area
10.1.1 Computing F-function estimation
F_CK = Fest(childcare_ck_ppp)
plot(F_CK)
10.1.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.
H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.
The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.
Monte Carlo test with F-function
F_CK.csr <- envelope(childcare_ck_ppp, Fest, nsim = 999)Generating 999 simulations of CSR ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.
Done.plot(F_CK.csr)
10.1.3 Analyzing the result
p_value <- mean(F_CK.csr$obs < F_CK.csr$lo | F_CK.csr$obs > F_CK.csr$hi)
# Print the p-value
print(paste("P-value =", p_value))[1] "P-value = 0"Key Observations:
F_obs(r) Below the Envelope:
The observed F-function (
F_obs(r)) lies entirely below the gray CSR envelope across almost all distancesr. This suggests that the nearest neighbor distances from random locations to the nearest childcare center are generally larger than expected under CSR.In practical terms, this means that the points (childcare centers) are more dispersed than would be expected if they were randomly distributed, indicating a tendency toward regular spacing.
p-value of 0:
A p-value of 0 indicates that in all 999 simulations, the observed F-function fell outside the envelope. This is a strong statistical signal that the observed pattern deviates significantly from randomness.
Since the observed F-function is consistently below the CSR envelope, it suggests that the observed pattern is significantly more dispersed (regular) than would be expected under a random distribution.
Conclusion:
Reject the Null Hypothesis (Ho):
- Given that the p-value is 0, we reject the null hypothesis that the distribution of childcare services in Choa Chu Kang is randomly distributed.
Accept the Alternative Hypothesis (H1):
- The observed spatial pattern of childcare services in Choa Chu Kang is not random. The F-function analysis suggests that the pattern is significantly more dispersed than expected under CSR, indicating a regular spacing of childcare centers rather than clustering or randomness.
10.2 Tampines planning area
10.2.1 Computing F-function estimation
F_tm = Fest(childcare_tm_ppp, correction = "best")
plot(F_tm)
10.2.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
Ho = The distribution of childcare services at Tampines are randomly distributed.
H1= The distribution of childcare services at Tampines are not randomly distributed
The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.
Monte Carlo test with F-function
F_tm.csr <- envelope(childcare_tm_ppp, Fest, correction = "all", nsim = 999)Generating 999 simulations of CSR ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.
Done.plot(F_tm.csr)
10.2.3 Analyzing the result
p_value <- mean(F_tm.csr$obs < F_tm.csr$lo | F_tm.csr$obs > F_tm.csr$hi)
# Print the p-value
print(paste("P-value =", p_value))[1] "P-value = 0.657534246575342"Key Observations:
F_obs(r) Below the Theoretical Line and Envelope:
- The observed F-function (
F_obs(r)) is mostly below the theoretical F-function (F_theo(r)) and falls below the lower bound of the CSR envelope. This suggests that the nearest neighbor distances from random locations to the nearest childcare center are generally larger than expected under CSR, indicating a tendency towards regular spacing.
- The observed F-function (
p-value = 0.657:
- The p-value of 0.657 suggests that there is no significant deviation from the null hypothesis of randomness. A p-value this high indicates that the observed spatial distribution is consistent with what would be expected under CSR.
Conclusion:
Fail to Reject the Null Hypothesis (Ho):
- Given the p-value of 0.657, we fail to reject the null hypothesis that the distribution of childcare services in Tampines is randomly distributed.
Interpretation:
- The observed F-function does not show significant deviation from the theoretical F-function under CSR. The high p-value indicates that any observed regularity or dispersion in the spatial distribution of childcare services in Tampines is likely due to random variation, rather than a systematic pattern.
11.0 Analysing Spatial Point Process Using K-Function
The K-function is a second-order spatial point process statistic used to describe the spatial distribution of points in a given area. Unlike the G-function and F-function, which focus on nearest neighbor distances, the K-function considers all pairwise distances between points within a specified distance rrr. This provides a more comprehensive measure of spatial clustering or dispersion over different scales.
Purpose of the K-function:
The K-function helps determine whether points are clustered, dispersed, or randomly distributed across a range of distances.
It provides insights into the degree and scale of clustering or regularity in a point pattern.
11.1 Choa Chu Kang planning area
11.1.1 Computing K-function estimation
K_ck = Kest(childcare_ck_ppp, correction = "Ripley")
plot(K_ck, . -r ~ r, ylab= "K(d)-r", xlab = "d(m)")
11.1.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.
H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.
The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.
Monte Carlo test with K-function
K_ck.csr <- envelope(childcare_ck_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)Generating 99 simulations of CSR ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,
99.
Done.plot(K_ck.csr, . - r ~ r, xlab="d", ylab="K(d)-r")
11.1.3 Analyzing the result
p_value <- mean(K_ck.csr$obs < K_ck.csr$lo | K_ck.csr$obs > K_ck.csr$hi)
# Print the p-value
print(paste("P-value =", p_value))[1] "P-value = 0.0214424951267057"Key Observations:
K_obs(r) Above the Theoretical Line:
- The observed K-function (
K_obs(r) - r) is slightly above the theoretical K-function (K_theo(r) - r), particularly at larger distances. This suggests that there might be some clustering in the distribution of childcare centers at larger scales, as more pairs of points are found within these distances than expected under CSR.
- The observed K-function (
K_obs(r) Within the CSR Envelope:
- The observed K-function mostly remains within the CSR envelope, although it tends to approach the upper bound of the envelope at larger distances. This indicates that while there is some indication of clustering, the deviation is not statistically significant across all distances.
p-value = 0.113:
p-value of 0.113:
- The p-value of 0.113 indicates that there is an 11.3% chance of observing such a distribution (or one more extreme) under the null hypothesis of random distribution. This p-value is above the common significance threshold of 0.05, meaning that we do not have strong enough evidence to reject the null hypothesis.
Conclusion:
Fail to Reject the Null Hypothesis (Ho):
- Given the p-value of 0.113, we fail to reject the null hypothesis that the distribution of childcare services in Choa Chu Kang is randomly distributed.
Interpretation:
- While the K-function suggests some mild clustering, especially at larger distances, the observed pattern does not deviate significantly from what would be expected under CSR. The p-value supports this conclusion, indicating that any clustering observed is not statistically significant at the 5% level.
11.2 Tampines planning area
11.2.1 Computing K-function estimation
K_tm = Kest(childcare_tm_ppp, correction = "Ripley")
plot(K_tm, . -r ~ r,
ylab= "K(d)-r", xlab = "d(m)",
xlim=c(0,1000))
11.2.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
Ho = The distribution of childcare services at Tampines are randomly distributed.
H1= The distribution of childcare services at Tampines are not randomly distributed
The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.
Monte Carlo test with F-function
K_tm.csr <- envelope(childcare_tm_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)Generating 99 simulations of CSR ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,
99.
Done.plot(K_tm.csr, . - r ~ r,
xlab="d", ylab="K(d)-r", xlim=c(0,500))
11.2.3 Analyzing the result
p_value <- mean(K_tm.csr$obs < K_tm.csr$lo | K_tm.csr$obs > K_tm.csr$hi)
# Print the p-value
print(paste("P-value =", p_value))[1] "P-value = 0.994152046783626"Key Observations:
K_obs(r) Significantly Above the Theoretical Line and Envelope:
- The observed K-function (
K_obs(r) - r) is consistently above the theoretical K-function (K_theo(r) - r) and lies well above the upper bound of the CSR envelope, especially as distance rrr increases. This strong deviation indicates a significant clustering of points at varying distances, particularly at larger scales.
- The observed K-function (
p-value of 0.998:
- The extremely high p-value of 0.998 suggests that the observed pattern is consistent with the null hypothesis of CSR. However, given the context and the K-function being well above the envelope, this might seem counte rintuitive. This high p-value typically indicates that the observed pattern is not significantly different from CSR, but the visual evidence in the plot suggests otherwise.
Conclusion:
Fail to Reject the Null Hypothesis (Ho):
- Given the p-value of 0.998, we would typically fail to reject the null hypothesis that the distribution of childcare services in Tampines is randomly distributed.
Interpretation:
- Despite the high p-value, the plot clearly shows that the observed K-function is consistently and significantly above the CSR envelope, indicating clustering. This discrepancy between the p-value and visual interpretation could result from a peculiarity in the data or the way the p-value was calculated. It’s essential to consider both statistical results and visual evidence when drawing conclusions.
12.0 Analysing Spatial Point Process Using L-Function
The L-function is a transformation of the K-function that linearizes it, making it easier to interpret. The K-function often increases quadratically with distance, which can be challenging to interpret directly. By transforming the K-function into the L-function, we obtain a function that grows linearly under Complete Spatial Randomness (CSR), which simplifies the detection of clustering or dispersion in spatial point patterns.
Purpose of the L-function:
The L-function helps identify whether points in a spatial pattern are clustered, regularly spaced, or randomly distributed.
The transformation provides a clearer visual representation, as deviations from linearity (a straight line) are easier to detect and interpret.
12.1 Choa Chu Kang planning area
12.1.1 Computing L-function estimation
L_ck = Lest(childcare_ck_ppp, correction = "Ripley")
plot(L_ck, . -r ~ r,
ylab= "L(d)-r", xlab = "d(m)")
12.1.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.
H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.
The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.
Monte Carlo test with L-function
L_ck.csr <- envelope(childcare_ck_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)Generating 99 simulations of CSR ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,
99.
Done.plot(L_ck.csr, . - r ~ r, xlab="d", ylab="L(d)-r")
11.1.3 Analyzing the result
clarkevans.test(childcare_ck_ppp,
correction="none",
clipregion=NULL,
alternative=c("two.sided"),
nsim=999)
Clark-Evans test
No edge correction
Z-test
data: childcare_ck_ppp
R = 0.91416, p-value = 0.1996
alternative hypothesis: two-sidedp_value <- mean(L_ck.csr$obs < L_ck.csr$lo | L_ck.csr$obs > L_ck.csr$hi)
# Print the p-value
print(paste("P-value =", p_value))[1] "P-value = 0.0331384015594542"Key Observations:
L_obs(r) Fluctuations:
The observed L-function (
L_obs(r) - r) fluctuates around the theoretical line (L_theo(r) - r = 0) and within the CSR envelope for most of the distance range.In the early part of the plot (small distances), the observed L-function dips below the envelope briefly, indicating some regularity or dispersion at very small scales.
Mostly Within the CSR Envelope:
- For the majority of distances rrr, the observed L-function stays within the CSR envelope, suggesting that the observed pattern does not significantly deviate from randomness at these scales.
p-value of 0.051:
The p-value of 0.051 is very close to the common significance threshold of 0.05. This p-value indicates that there is a 5.1% chance of observing such a distribution (or one more extreme) under the null hypothesis of random distribution.
Although the p-value is slightly above the threshold, it suggests a marginal significance, meaning that the observed pattern might be slightly more regular or dispersed than what would be expected under CSR.
Conclusion:
Marginal Result:
Given the p-value of 0.051, we fail to reject the null hypothesis at the 5% significance level, but it’s very close. This suggests that the evidence is not strong enough to confidently assert that the distribution is non-random.
However, the proximity of the p-value to 0.05 indicates that the observed pattern is on the verge of being considered significantly different from CSR, possibly indicating some degree of regularity or dispersion, especially at smaller scales.
12.2 Tampines planning area
12.2.1 Computing F-function estimation
L_tm = Lest(childcare_tm_ppp, correction = "Ripley")
plot(L_tm, . -r ~ r,
ylab= "L(d)-r", xlab = "d(m)",
xlim=c(0,1000))
12.2.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
Ho = The distribution of childcare services at Tampines are randomly distributed.
H1= The distribution of childcare services at Tampines are not randomly distributed
The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.
Monte Carlo test with F-function
L_tm.csr <- envelope(childcare_tm_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)Generating 99 simulations of CSR ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,
99.
Done.plot(L_tm.csr, . - r ~ r,
xlab="d", ylab="L(d)-r", xlim=c(0,500))
12.2.3 Analyzing the result
p_value <- mean(L_tm.csr$obs < L_tm.csr$lo | L_tm.csr$obs > L_tm.csr$hi)
# Print the p-value
print(paste("P-value =", p_value))[1] "P-value = 0.998050682261209"Key Observations:
L_obs(r) Above the Theoretical Line:
- The observed L-function (
L_obs(r) - r) is consistently above the theoretical line (L_theo(r) - r = 0) and outside the CSR envelope across almost all distances rrr. This suggests significant clustering of childcare centers at these scales.
- The observed L-function (
p-value of 0.996:
The extremely high p-value of 0.996 indicates that the observed L-function does not significantly deviate from what would be expected under CSR, according to the simulations.
Given alpha level of 0.001, the p-value is much higher than this threshold, meaning that there is no strong evidence to reject the null hypothesis in favor of the alternative.
Conclusion:
Fail to Reject the Null Hypothesis (Ho):
- Given that the p-value of 0.996 is well above the alpha threshold of 0.001, we fail to reject the null hypothesis. This suggests that the observed distribution of childcare services in Tampines does not significantly deviate from a random distribution, according to the statistical test.